arXiv research

Study rules out exotic $S^4$ and $\# n \mathbb{CP}^2$ construction using zero surgery homeomorphisms.

Paper explores using 0-surgery to find exotic 4-manifolds.

We use topological surgery in dimension four to give sufficient conditions for the zero framed surgery manifold of a 3-component link to be homology cobordant to the 3-torus, which arises from zero framed surgery on the Borromean rings, via a topological homology cobordism preserving the homotopy classes of the meridia…

Dehn surgery homeomorphic pairs contradict a conjecture.

Extends knot surgery to exotic four-manifolds.

Maps can transform non-homeomorphic manifolds into the same space.

The paper proves a theorem about shared Dehn surgeries between knots.

We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floe…

New insights into cosmetic surgeries using Heegaard Floer homology.

Two Dehn surgeries on a knot are called cosmetic if they yield homeomorphic manifolds. For a null-homologous knot with certain conditions on the Thurston norm of the ambient manifold, if the knot admits cosmetic surgeries, then the surgery coefficients are equal up to sign.

Paper restricts chirally cosmetic surgeries on knots.

Surface corks modify 4-manifold structures without changing their homeomorphism type.

New proof for a knot type not admitting certain surgeries.

We study chirally cosmetic surgeries, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3-manifolds with opposite orientations. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredients are the original and the $SL(2,\mathbb{C})$ version of Casson invariant…

The study finds bounds on characterizing slopes for all knots.

Two Dehn surgeries on a knot are called {\it purely cosmetic}, if they yield manifolds that are homeomorphic as oriented manifolds. Suppose there exist purely cosmetic surgeries on a knot in $S^3$, we show that the two surgery slopes must be the opposite of each other. One ingredient of our proof is a Dehn surgery form…

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for cable knots.

3-braid knots can't have purely cosmetic surgeries.

We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite knot or a hyperbolic knot, except that both cannot be satellite knots simultaneo…

We study collections of curves in generic position on a closed surface whose complement consists of one disk only, up to orientation-preserving homeomorphism of the surface. We define a surgery operation on the set of such collections and prove that any two of them can be connected by a sequence of such surgeries.

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for connected sums of knots by analysing the JSJ-structures.

A theorem of Kirby states that two framed links in the 3-sphere produce orientation-preserving homeomorphic results of surgery if they are related by a sequence of stabilization and handle-slide moves. The purpose of the present paper is twofold: First, we give a sufficient condition for a sequence of handle-slides on …

In this paper, given a knot K, for any integer m we construct a new surface Sigma_K(m) from a smoothly embedded surface Sigma in a smooth 4-manifold X by performing a surgery on Sigma. This surgery is based on a modification of the `rim surgery' which was introduced by Fintushel and Stern, by doing additional twist spi…

Kawauchi defined a group structure on the set of homology $S^1$\times$S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. It is natural to ask whether the zero-surgery homomorphism is injecti…

Special knots with many twists have no certain type of surgery.

New Heegaard Floer homology findings block chirally cosmetic surgeries.

The paper classifies bundles over complex projective plane.

Using the correction terms in Heegaard Floer homology, we prove that if a knot in $S^3$ admits a positive integral $\mathbf{T}$-, $\mathbf{O}$- or $\mathbf{I}$-type surgery, it must have the same knot Floer homology as one of the knots given in our complete list, and the resulting manifold is orientation-preservingly h…

By considering non-orientable surfaces in the surgered manifolds, we show that the 10/3- and -10/3-Dehn surgeries on the 2-bridge knot $9_{27} = S(49,19)$ are not cosmetic, i.e., they give mutually non-homeomorphic manifolds. The knot is unknown to have no cosmetic surgeries by previously known results; in particular, …

Study shows most odd pretzel knots don't allow chirally cosmetic surgeries.

Study of knots sharing 0-surgeries, classifying and computing their properties.

The study proves large alternating Montesinos knots cannot have purely cosmetic surgeries.

This paper concerns the Dehn surgery construction, especially those Dehn surgeries leaving the manifold unchanged. In particular, we describe an oriented 1-cusped hyperbolic 3-manifold X with a pair of slopes r_1, r_2 such that the Dehn filled manifolds X(r_1), X(r_2) are oppositely oriented copies of the lens space L(…

New method finds infinitely many knots in 3-manifolds.

For 5 <= k <= 8 we show that the infinite family of exotic smooth structures on CP^2# k(-CP^2) can be achieved by 1/n - surgeries on a single embedded nullhomologous torus in a manifold R_k which is homeomorphic to CP^2# k(-CP^2).

We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V_K''(1)\neq 0$ or $V_K'''(1)\neq 0$. As an application, we verify the cosmetic surgery conjecture for all knots with no more than $11$ crossings except for three $10$-crossing knots and five $11$-crossin…

Study finds infinitely many surface bundle types with zero signature.

Fintushel-Stern's knot surgery gave many pairs of exotic manifolds, which are homeomorphic but non-diffeomorphic. We show that if an elliptic fibration has two parallel, oppositely oriented vanishing circles (for example $S^2\times S^2$ or Matsumoto's $S^4$), then the knot surgery gives rise to standard manifolds. The …

New method proves cosmetic surgery conjecture for certain knots.

The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has a negative answer. Namely, we give examples of knots whose zero-framed surgeries…

We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, i.e., no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument, based on a recent result by Hanselman, uses several invariants of knots or 3-manifolds; f…

The paper classifies 4-manifolds with given boundaries.

In the paper \cite{wall_1}, C.T.C. Wall proved that two smooth closed simply connected 4-manifolds which are homeomorphic are in fact stably diffeomorphic. We prove a similar result which states that two smooth closed 4-manifolds satisfying certain properties are stably diffeomorphic if and only if their signatures agr…

We give a complete description of exceptional surgeries on pretzel knots of type $(-2, p, p)$ with $p \ge 5$. It is known that such a knot admits a unique toroidal surgery yielding a toroidal manifold with a unique incompressible torus. By cutting along the torus, we obtain two connected components, one of which is a t…

Let K be a knot in the 3-sphere. A slope p/q is said to be characterising for K if whenever p/q surgery on K is homeomorphic, via an orientation-preserving homeomorphism, to p/q surgery on another knot K' in the 3-sphere, then K and K' are isotopic. It was an old conjecture of Gordon, proved by Kronheimer, Mrowka, Ozsv…

New slopes identified for torus knots, improving previous results.

We show that for any nontrivial knot in $S^3$, there is an open interval containing zero such that a Dehn surgery on any slope in this interval yields a 3-manifold with taut foliations. This generalizes a theorem of Gabai on zero frame surgery.

We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically sl…